| In this dissertation, we describe the structure of two important classes of invariant positive definite kernels on the free semigroup on finitely many generators. Also, we study their orthogonal polynomials. The structure results are of a recursive nature, but we also establish a connection with the combinatorics of lattice paths of Motzkin type. In addition, we describe a general procedure for the construction of analogues in several non-commuting variables of all classical orthogonal polynomials. |
